Practice Power indices and surds - quantitative aptitude Online Quiz (set-1) For All Competitive Exams

$√{ - √{3} + √{3 + 8√{7 + 4√{3}}}}$

= $√{ - √{3} + √{3 + 8√{4 + 3 + 2 × 2 × √{3}}}}$

= $√{ - √{3} + √{3 + 8√{(2)^2 + (√3)^2 + 2 × 2 × √{3}}}}$

= $√{ - √{3} + √{3 + 8√{(2 × √{3})^2}}}$

= $√{ - √{3} + √{3 + 8(2 × √{3})}}$

= $√{ - √{3} + √{3 + 16 +8√{3}}}$

= $√{-√{3} + √{(√3)^2 + (4)^2 + 2 × 4 √{3}}}$

=$√{-√{3} + √{(4 + √3)^2}}$

= $√{-√{3} + 4 +√{3}} = √4$ = 2

$(16)^{0.16} × (16)^{0.04} × (2)^{0.2}$

= $(2^4)^{0.16} × (2^4)^{0.04} × (2)^{0.2}$

= $2^{0.64} × 2^{0.16} × 2^{0.2}$

= $(2)^{0.64+0.16+0.2}$ = 2

? = $(√8 - √4 - √2)$

= $(2√2 - 2 - √2) = √2 - 2$

$(16^{3/2} + 16^{-3/2})$

= $(4^2)^{3/2} + 1/(16)^{3/2}$

= $4^{2×3/2} + 1/4^{2×3/2} = 4^3 + 1/4^3$

= $64 + 1/64 = {4096 + 1}/64 = 4097/64$

$2/{√7 + √5} + 7/{√{12} - √5} - 5/{√{12} - √7}$

First term = $2/{√7 + √5}$

= ${2 ×(√7 - √5)}/{(√7 + √5)(√7 - √5)}$

= ${2(√7 - √5)}/{7-5} = √7 -√5$

Second term = $7/{√{12} - √5}$

= ${7(√{12}+√5)}/{(√{12}-√{5})(√{12}+√5)}$

=${7(√{12}+√5)}/{12-5}$

= ${7(√{12} +√{5})}/{7} = √{12}+√5$

Third term = $5/{√{12}-√7}$

= ${5(√{12}+√7)}/{(√{12}-√7)(√{12}+√7)}$

= ${5(√{12}+√7)}/{12- 7} = √{12} +√7$

Expression

= $(√7 - √5) + (√{12} +√5) - (√{12}+√7)$

= $√7 -√5 +√{12} +√5 - √{12} -√7 = 0$

x = $√{12+√{12+√{12 +...}}}$

On squaring both sides,

$x^2 = 12 + √{12+√{12+√{12 +...}}}$

$x^2$ = 12 + x

$x^2$ - x - 12 = 0

$x^2$ - 4x + 3x - 12 = 0

x (x - 4) + 3 (x - 4) = 0

(x - 4) (x + 3) = 0

x = 4 because x ≠ - 3

${√{10+√{ 25+√{ 108+√{ 154+√{ 225}}}}}}/√^3{8}$= ?

?=${√{10+√{ 25+√{ 108+√{ 154+15}}}}}/√^3{2×2×2}$

=${√{10+√{ 25+√{ 108+√{169}}}}}/2$

=${√{10+√{ 25+√{ 108+13}}}}/2$

=${√{10+√{ 25+√{121}}}}/2$

=${√{10+√{ 25+11}}}/2$

=${√{10+√{36}}}/2={√{10+6}}/2$

=${√{16}}/2=4/2=2$

Let x = $√{2+√{2+√{2 +...}}}$

On squaring both sides

$x^2 = 2+√{2+√{2+√{2 +...}}}$

$x^2$ = 2 + x

$x^2$ - x - 2 = 0

$x^2$ - 2x + x - 2 = 0

x (x - 2) + 1 (x - 2) = 0

(x - 2) (x + 1) = 0

x = 2 or - 1

But sum of positive numbers can't be negative.

x = 2

Let x = $√{30+√{30+√{30+...}}}$

On squaring both sides,

$x^2 = 30+√{30+√{30+√{30+...}}}$

$x^2$ = 30 + x

$x^2$ - x - 30 = 0

$x^2$ - 6x + 5x - 30 = 0

x (x - 6) + 5 (x - 6) = 0

(x - 6) (x + 5) = 0

x = 6 because x ≠ - 5

Using Rule 25

$√{30+√{30+√{30+...}}}$ = 6

It is because

30 = 5 × 6 = n (n +1)

Let, x = $√{6+√{6+√{6 +...∞}}}$

On squaring,

$x^2 = 6 + √{6+√{6+√{6 +...∞}}}$

$x^2$ = 6 + x

$x^2$ - x - 6 = 0

$x^2$ - 3x + 2x - 6 = 0

x (x - 3) + 2 (x - 3) = 0

(x + 2) (x - 3) = 0

x = 3 because x ≠ - 2

Using Rule 25

$√{6+√{6+√{6 +...}}}$ = 3

It is because, 6 = 2 × 3 =n (n+1)

${4 +3√3}/{√{7 + 4 √3}}$

${4 +3√3}/{√{7 + 4 √3}}={4+3√3}/{√{4+3+2×2×√3}}$

=${4+3√3}/{√(2+√3)^2}={4+3√3}/{2+√3}$

${(4+3√3)(2-√3)}/{(2+√3)(2-√3)}$

=$8-4√3+6√3-9$

=$2√3-1=2×1.732-1$

= 3.464 - 1 = 2.464

Expression=$1/{√5 +√3}$

=$1/{√5 +√3}×{√5 -√3}/{√5 -√3}$

(Rationalising the denominator)

=${√5 -√3}/{5-3}={2.236-1.732}/2$

=$0.504/2=0.252$

Expression

=$1/{√2 + 1}={√2 - 1}/{(√2 + 1)(√2 - 1)}$

=${√2-1}/{2-1}=√2-1$

= 1.414 - 1 = 0.414

Expression

=${3 +√6}/{5√3 - 2√12- √32+√ 50}$

=${3 +√6}/{5√3 - 2√{2×2×3}- √{2×2×2×2×2}+√{2×5×5}}$

=${3 +√6}/{5√3 - 4√{3}- 4√{2}+5√{2}}$

=${3 +√6}/{√3 +√2}={(3+√6)(√3-√2)}/{(√3+√2)(√3-√2)}$

[On rationalising the denominator]

=${3√3+√18-3√2-√12}/{3-2}$

=$3√3+3√2-3√2-2√3$

=$√3=1.732$

$√33$ = 5.745(Given)

$√{3/11}=√{{3×11}/{11×11}}=√33/11$

=${5.745}/11 ≈0.5223$

$√^3{2}$ or $√3$

$√^3{2} =2^{1/3}=2^{2/6}=√^6{4}$

$√{3} =3^{1/2}=3^{3/6}=√^6{27}$

$16^{3/4} = (4^2)^{3/4}$

= $4^{2×3/4} = 4^{3/2} = 2^{2×3/2} = 2^3 = 8$

Let x = $√{3√{3√{3...}}}$

Squaring both sides,

$x^2 = 3√{3√{3√{3...}}}$ = 3x

$x^2$ - 3x = 0

x (x - 3) = 0

x = 3 because x ≠ 0

$√{3+√{3+√{3+...∞}}}$ = 3

It is because, here

n = ∞ and x =3

$√{3+√{3+√{3+...∞}}}=3^(1-1/{3∞})$

= $3^(1 - 0)$ [${something}/∞ = 0$] = 3

Expression

=$√{9 + 2√{16} +√^3{ 512}}$

=$√{9 + 2√{4×4} +√^3{8×8×8}}$

=$√{9 + 2×4+8}$

=$√25$ =5

$3√2, 3√7, 6√5, 2√20$

$3√2=3×1.4=4.2$

$3√7=3×2.6=7.8$

$6√5=6×2.2=13.2$

$2√20=2×4.5=9$